Modulating a digital signal with narrow spectrum and substantially constant envelope

ABSTRACT

A transmission signal (S) resulting from modulation of a digital data signal by a modulation function dependent on time t, the data signal being formed of a series of bits (b k ) each identified by its rank k and having a duration T, the transmission signal (S) consisting of a summation indexed to the rank k of the product of the complex constant j to the power k, of the modulation function (h(t−kT) and of an input signal. The input signal F(k) being a function of the data signal (b k ), the modulation function h(t−kT) is a Gaussian function of time t.

RELATED APPLICATION

This application is the national filing of international applicationnumber PCT/FR98/02812, which claims the priority of FR 97/16454 filedDec. 22, 1997.

BACKGROUND OF THE INVENTION

The present invention relates to a technique for modulating a digitalsignal.

It therefore applies to the transmission art, in particular the radiotransmission art.

A typical application is aimed at radio systems, in particular“broadband” systems. These systems are naturally designed to offer ahigh capacity and it is therefore necessary to use a modulationtechnique that offers high spectral efficiency, which amounts to sayingthat the spectrum of a given channel must be as narrow as possibleconsistent with its technical specifications.

Moreover, constant envelope modulation is generally employed, whichminimizes the complexity of transmitters. If the signal featuresvariations of relatively large amplitude, the amplification stages mustbe perfectly linear, especially the power amplifier.

It nevertheless appears to be the case that all forms of constantenvelope modulation known in the art until now have a spectrum whichfeatures a side lobe.

If it is within the spectrum of a neighboring channel, the side lobeincreases the level of interference in that neighboring channel.

The Gaussian minimum shift keying (GMSK) modulation technique employedin the GSM has a −40 dBc side lobe at 200 kHz from the main lobe. Thespacing between two adjacent channels is also 200 kHz. Obviously thisreduces spectral efficiency.

The spectrum of quadrature amplitude modulation (QAM) has no side lobeif an appropriate filter is used.

However, these modulation techniques cause strong variations in theamplitude of the modulated signal. As mentioned above, it is necessaryto use more complex and therefore more cost amplifiers in this case.

SUMMARY OF THE INVENTION

The object of the present invention is therefore to provide a modulationtechnique which offers a spectrum with no side lobe but which still hasa practically constant envelope.

The invention therefore provides a transmission signal resulting frommodulation of a digital data signal by a modulation function dependenton time t, the data signal being formed of a series of bits b_(k) eachidentified by its rank k and having a duration T, the transmissionsignal (S) consisting of a summation indexed to the rank k of theproduct of the complex constant i to the power k, of the modulationfunction h(t−kT) and of an input signal. According to the invention,since the input signal is a function of the data signal, the modulationfunction is a Gaussian function of time t.

The modulation function is advantageously defined by the followingequation, in which the parameter σ is a form factor which determines thespreading of a bit:${h(t)} = {\frac{1}{\sigma\quad T\sqrt{2\pi}}{\mathbb{e}}^{- \frac{t^{2}}{2\sigma^{2}T^{2}}}}$

The simplest embodiment of the invention defines the input signal asequal to the data signal.

However, if this solution is adopted, the transmission signal is stillsubject to amplitude variations which, although minimal, still imposesome constraints on the amplifiers of the transmitter.

Accordingly, the input signal E(k) preferably has the value:${F(k)} = {\sum\limits_{n = 0}^{N}\quad{a^{n}B_{k}^{n}}}$

-   -   where:    -   N is a positive natural integer,    -   a is a positive correction constant,    -   the polynomial B_(k) ^(n) is defined as follows:    -   where:        $B_{k}^{n} = {\sum\limits_{1 = i}^{L}\quad\left( {\prod\limits_{i = 0}^{2\quad M}\quad b_{k + p_{1,i}}} \right)}$    -   a family (F₁) of relative integers P_(l,i) is constructed so        that there is a natural integer M    -   which satisfies the following equations:        ${\sum\limits_{i = 0}^{M}\quad p_{1,{2i}}} = {\sum\limits_{i = 1}^{M}\quad p_{1,{{2i} - 1}}}$        ${{\sum\limits_{i = 0}^{M}\quad p_{1,{2i}}^{2}} - {\sum\limits_{i = 1}^{M}\quad p_{1,{{2i} - 1}}^{2}}} = {2\quad n}$    -   p_(l,i)<p_(l,i+1) (for all i)    -   L represents the total number of these families.

For example, the correction constant has the value${\mathbb{e}}^{- \frac{1}{\sigma^{2}}}.$

The invention also relates to a modulator for producing the transmissionsignal.

In a preferred embodiment of the invention, the modulator comprises adigital processor which receives said input signal and produces the realpart and the imaginary part of the transmission signal, a first mixerwhich multiplies the real part by a carrier, a phase-shifter whichreceives the carrier and phase-shifts it by π/2, a second mixer whichmultiplies the imaginary part by the output signal of the phase-shifter,and an adder which sums the output signals of the two mixers.

When the correction level is greater than zero the digital processorincludes a first module for producing the polynomials B_(k) ^(n).

The digital processor further comprises a second module for producingdigital samples of the transmission signal, four samples E_(rk+1) beingassociated with the bit b_(k) for i varying from 0 to 3 and having thevalue:$E_{{4k} + i} = {\sum\limits_{q = k}^{k - 5}\quad{j^{q} \cdot \left( {b_{q} + {\frac{1}{8} \cdot B_{q}^{1}} + {\frac{1}{64} \cdot B_{q}^{2}}} \right) \cdot h_{{4{({k - q})}} + i}}}$

The invention also provides a demodulator for reproducing the datasignal from the transmission signal.

The demodulator preferably comprises a baseband transposition unit whichreceives a signal which has been modulated by the modulation function, acomplex multiplier which multiplies the output signal of thetransposition unit by the expression $e^{{- j}\frac{\pi\quad i}{2t}},$a convolution operator which convolutes the output signal of the complexmultiplier and the modulation function, and a decision unit whichreproduces the data signal as a function of the sign of the real part ofthe result of the convolution.

The transposition unit is routinely a Hilbert filter.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will now emerge in more detail from the followingdescription of embodiments of the invention, which description is givenby way of illustrative example only and with reference to theaccompanying diagrammatic drawings, in which:

FIG. 1 shows a modulator according to the invention,

FIG. 2 shows a first module of one embodiment of the modulator,

FIG. 3 shows a second module of the same embodiment of the modulator,and

FIG. 4 shows a demodulator according to the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

It is therefore a question of modulating a digital data signal whichconsists of a stream of bits b_(k) taking the value +1 or −1.

The following expression is known in the art for a modulatedtransmission signal S with various types of modulation:

-   -   where:        $S = {\sum\limits_{k}{j^{k} \cdot {h\left( {t - {kT}} \right)} \cdot b_{k}}}$    -   k is the index of the current bit b_(k),    -   j is the complex constant such that j²=−1,    -   T is the duration of a bit,    -   t represents time, and    -   h is the modulation function.

If h is a rectangular function, the modulation is referred to as “offsetQPSK”.

According to the invention, h is a Gaussian function which takes thefollowing form, for example:${h(t)} = {\frac{1}{\sigma\quad T\sqrt{2\pi}}{\mathbb{e}}^{- \frac{t^{2}}{2\sigma^{2}T^{2}}}}$

The parameter σ is a form factor which determines the spreading of abit.

If B denotes the half-bandwidth of the spectrum at the 3 dB points, andusing the abbreviation ln for natural logarithms, the following equationapplies: ${B \cdot T} = \frac{\sqrt{\ln\quad 2}}{2\quad{\pi\sigma}}$

Adopting a Gaussian modulation function eliminates the side lobe presentin constant envelope modulation spectra.

Note however that the modulated signal is still subject to amplitudevariations, although they are significantly reduced compared to QAM.

Accordingly, and in accordance with another aspect of the invention, acorrection term C is added to the modulated signal as defined above: ewith:$S = {{\sum\limits_{k}{j^{k} \cdot {h\left( {t - {kT}} \right)} \cdot b_{k}}} + C}$

-   -   where:        $C = {\sum\limits_{k}{j^{k} \cdot {h\left( {t - {kT}} \right)} \cdot \left\lbrack {\sum\limits_{n = 1}^{N}\quad{a^{n}B_{k}^{n}}} \right\rbrack}}$    -   N is an integer representing a correction level, and    -   a is a constant with the value $e^{- \frac{1}{\sigma^{2}}}.$

The polynomials B_(k) are constructed in the following manner. For agiven value of n, all families of relative integers P_(l,i) are foundsuch that there is a natural integer M satisfying the followingequations: $\begin{matrix}{{\sum\limits_{i = 0}^{M}\quad p_{1,{2i}}} = {\sum\limits_{i = 1}^{M}\quad p_{1,{{2i} - 1}}}} & (1) \\{{{\sum\limits_{i = 0}^{M}\quad p_{1,{2i}}^{2}} - {\sum\limits_{i = 1}^{M}\quad p_{1,{{2i} - 1}}^{2}}} = {2\quad n}} & (2)\end{matrix}$  p _(l,i) <p _(l i+1) (for all i)  (3)

The limit values of p_(l,i) must be found. They are p_(1,0) andp_(1,2M).

From equation (1) it follows that p_(1,0) is negative or zero:$p_{1,0} = {\sum\limits_{i = 1}^{M}\quad\left( {p_{1,{{2i} - 1}} - p_{1,{2i}}} \right)}$

From equation (3), the above expression is negative or zero.

Similarly, the term P_(1,2M) is positive or zero.

Equation (1) can be written:$p_{1,{2M}} = {\sum\limits_{i = 1}^{M}\quad\left( {p_{1,{{2i} - 1}} - p_{1,{{2i} - 2}}} \right)}$

From equation (3), this expression is positive or zero.

Because the relative integers p_(l,i) constitute a strictly increasingseries, there is only one natural integer z for which the productp_(l,x)·p_(l,z−1) is negative or zero.

If z is even, equation (2) can be written:${{\sum\limits_{i = {\frac{z}{2} + 1}}^{M}\quad\left( {p_{1,{2i}}^{2} - p_{1,{{2i} - 1}}^{2}} \right)} + p_{1,z}^{2} + {\sum\limits_{i = 0}^{\frac{z}{2} - 1}\quad\left( {p_{1,{2i}}^{2} - p_{1,{{2i} + 1}}^{2}} \right)}} = {2n}$

If z is odd, equation (2) can be written:${{\sum\limits_{i = \frac{z + 1}{2}}^{M}\quad\left( {p_{1,{2i}}^{2} - p_{1,{{2i} - 1}}^{2}} \right)} + p_{1,{z - 1}}^{2} + {\sum\limits_{i = 0}^{\frac{z - 3}{2}}\quad\left( {p_{1,{2i}}^{2} - p_{1,{{2i} + 1}}^{2}} \right)}} = {2n}$

In the above two expressions, the left-hand member of the equation takesthe form of a sum of positive terms, which implies that each of thoseterms is at most equal to 2n.

Accordingly:

-   -   if z=2M:        p_(1,2M)<√{square root over (2n)}    -   if z<2M:        p_(1,2M) ² −p _(1,2M−1) ²≦2n        (p ^(1,2M) −p _(1,2M−1))(p _(1,2M) −p _(1,2M−1))≦2n    -   letting (p_(1,2M) −p _(1,2M−1))=a, with 1≦a≦2n        ${{{2p_{1,{2M}}} - a} \leq \frac{2n}{a}},{p_{1,{2M}} \leq {\frac{n}{a} + \frac{a}{2}}}$

It can easily be shown that (n/a+a/2)≦(n+1/2) when a is between 1 and2n.

It follows that p_(1,2M) is less than or equal to n. It can be shown inthe same way that p_(1,0) is greater than or equal to −n.

It follows from the above that the set of relative integer familiesp_(l,i) is a finite set.

For a given value of n, now consider the first family p_(l,i), which isobtained for l=1. That family is constructed starting from p_(1,0)=−n,after which the series of relative integers p_(1,1), . . . , p_(1,2M)which satisfies equations (1), (2) and (3) is found empirically.

For example, if n=1, there is only one family F₁={p_(1,0), p_(1,1),p_(1,2)}={−1, 0, +1}.

If n is greater than 1, all the families are found in the same manner bysuccessively incrementing all p_(l,i). In this case, l varies from 1 toL.

For the first values of n, those families are: $\begin{matrix}{{{- n} = 2},{F_{1} = {\left\{ {p_{l,0},p_{l,1},p_{l,2}} \right\} = \left\{ {{- 2},{- 1},{+ 1}} \right\}}}} \\{F_{2} = {\left\{ {p_{2,0},p_{2,1},p_{2,2}} \right\} = \left\{ {{- 1},{+ 1},{+ 2}} \right\}}}\end{matrix}$ $\begin{matrix}{{{- n} = 3},{F_{1} = {\left\{ {p_{l,0},p_{l,1},p_{l,2}} \right\} = \left\{ {{- 3},{- 2},{+ 1}} \right\}}}} \\{F_{2} = {\left\{ {p_{2,0},p_{2,1},p_{2,2},p_{2,3},p_{2,4}} \right\} = \left\{ {{- 2},{- 1},0,{+ 1},{+ 2}} \right\}}} \\{F_{3} = {\left\{ {p_{3,0},p_{3,1},p_{3,2}} \right\} = \left\{ {{- 1},{+ 2},{+ 3}} \right\}}}\end{matrix}$ $\begin{matrix}{{{- n} = 4},{F_{1} = {\left\{ {p_{l,0},p_{l,1},p_{l,2}} \right\} = \left\{ {{- 4},{- 3},{+ 1}} \right\}}}} \\{F_{2} = {\left\{ {p_{2,0},p_{2,1},p_{2,2},p_{2,3},p_{2,4}} \right\} = \left\{ {{- 3},{- 2},0,{+ 1},{+ 2}} \right\}}} \\{F_{3} = {\left\{ {p_{3,0},p_{3,1},p_{3,2}} \right\} = \left\{ {{- 2},{+ 0},{+ 2}} \right\}}} \\{F_{4} = {\left\{ {p_{4,0},p_{4,1},p_{4,2},p_{4,3},p_{4,4}} \right\} = \left\{ {{- 2},{- 1},0,{+ 2},{+ 3}} \right\}}} \\{F_{5} = {\left\{ {p_{5,0},p_{5,1},p_{5,2}} \right\} = \left\{ {{- 1},{+ 3},{+ 4}} \right\}}}\end{matrix}$

Finally, the polynomial B_(k) ^(n) is obtained from the followingexpression:$B_{k}^{n} = {\sum\limits_{1 = i}^{L}\quad\left( {\prod\limits_{i = 0}^{2\quad M}\quad b_{k + p_{1,i}}} \right)}$

Returning to the previous examples:B _(k) ¹ =b _(k−1) .b _(k) .b _(k+1)B _(k) ² =b _(k−2) .b _(k−1) .b _(k+1) +b _(k−1) .b _(k+1) .b _(k+2 3)B _(k) ³ =b _(k−3) .b _(k−2) .b _(k+1) +b _(k−2) .b _(k−1) .b _(k) .b_(k+1) .b _(k+2) +b _(k−1) .b _(k+2) .b _(k+3)B _(k) ⁴ =b _(k−4) .b _(k−3) .b _(k+1) +b _(k−3) .b _(k−2) .b _(k) .b_(k+1) .b _(k+2) +b _(k−2) .b _(k) .b _(k+) ₂ +b _(k−2) .b _(k−1) .b_(k) .b _(k+2) .b _(k+3) +b _(k−1) .b _(k+3) .b _(k+4)

Returning to the equation of the modulated signal S:$S = {{\sum\limits_{k}{j^{k} \cdot {h\left( {t - {kT}} \right)} \cdot b_{k}}} + {\sum\limits_{k}{j^{k} \cdot {h\left( {t - {kT}} \right)} \cdot \left\lbrack {\sum\limits_{n = 1}^{N}\quad{a^{n}B_{k}^{n}}} \right\rbrack}}}$

Letting B_(k) ⁰=b_(k), the signal S can be written:$S = {\sum\limits_{k}{j^{k} \cdot {h\left( {t - {kT}} \right)} \cdot \left\lbrack {\sum\limits_{n = 0}^{N}\quad{a^{n}B_{k}^{n}}} \right\rbrack}}$

It is therefore possible to define an input signal${F(k)} = {\sum\limits_{n = 0}^{N}\quad{a^{n}B_{k}^{n}}}$

If N=0, this is the simplest embodiment of the invention. The greaterthe value of N, the more limited are the amplitude variations of themodulated signal S.

Note that the spectrum of this signal is independent of N. Its value is:[H _(σ)(f)]² =e ^(−(2πσfT)) ²

The invention naturally relates to a modulator for producing the signalS modulated onto a carrier. Although the implementation of a modulatorso specified will be obvious to the skilled person, one of many exampleswill now be described.

Referring to FIG. 1, the modulator compromises a digital processor 1which receives the bits b_(k) to produce the real part I and theimaginary part Q of the modulated signal S:S=I+jQ.

It also comprises a first mixer 4 for multiplying the real part I by thecarrier C and a second mixer 5 for multiplying the imaginary part Q bythe carrier phase-shifted by π/2. To this end, a phase-shifter 3receives the carrier and feeds it to the second mixer 5.

It also comprises an adder 6 for summing the output signals of the twomixers 4,5.

Finally, the modulator includes a time base 2 which supplies the clocksignal Ck to the digital processor 1 and the carrier to the first mixer4 and to the phase-shifter 3.

It works for the most varied values of the various constants and inparticular with a correction level N equal to 0. However, to obtain goodperformance, and to facilitate the task of the processor 1, thefollowing values are given by way of example:

-   -   form factor: ${\sigma = \frac{1}{\sqrt{3\ln\quad 2}}};$    -    because of this, the constant        $a = {\mathbb{e}}^{- \frac{1}{\sigma^{2}}}$    -    has the value ⅛, which enables multiplication by a means of a        shift of three bits to the right,    -   correction level N=2,    -   value of bits b_(k): +1 or −1,    -   modulated signal S expressed on 12 bits,    -   oversampling factor: 4.

The modulated signal S is therefore a series of digital samples producedat the rate of four per bit period T.

The modulation function h(t) is also represented by a series of positivenumbers h_(q) on 11 bits. An appropriate scale factor is chosen so thatthe modulated signal S can be coded on 12 bits:

-   -   (h_(q))_(0≦q≦11)={0,1,5,17,47,116,253,485,816,1205,1563,1780}

The function h(t) is even so that for any c from 0 to 11 h_(23−q)=h_(q).Given the scale factor adopted, h_(q) is zero for q<0 or q>23: thefunction is memorized for −3T<t<3T.

Because of the oversampling, it is possible to set q=4.k+i, for ivarying from 0 to 3; in other words, k is the integer part of q/4.

Referring to FIG. 2, and by the way of example, the processor 1comprises a first module for calculating the expressions B_(k) ¹ andB_(k) ². Here the corresponding calculations are performed by means ofshift register which contains the bits b_(k+2) to b_(k−2) at a referencetime. B_(k) ¹ is obtained by a first multiplier 7 which forms theproduct of the bits b_(k−1), b_(k) and b_(k+1). To obtain B_(k) ², asecond multiplier 8 forms the product of the bits b_(k−2), b_(k−1) andb_(k+1), a third multiplier 9 forms the product of the bits b_(k−1),b_(k+1), and b_(k+2), and an adder 10 sums the outputs of the second andthird multipliers 8 and 9.

The processor 1 also includes a second module, shown in FIG. 3, whichcalculates the digital samples of the modulated signal S by filteringthe oversampled input signals using a filter with impulse response h(t).The four samples E_(4k+1) associated with the bit b_(k) for i varyingfrom 0 to 3 therefore have the values:$E_{{4k} + i} = {\sum\limits_{q = k}^{k - 5}\quad{j^{q} \cdot \left( {b_{q} + {\frac{1}{8} \cdot B_{q}^{1}} + {\frac{1}{64} \cdot B_{q}^{2}}} \right) \cdot h_{{4{({k - q})}} + i}}}$

The above expression can be written:E _(4k+i) =E _(ki) ⁰ +E _(ki) ¹ +E _(ki) ²

-   -   where: $\begin{matrix}        {E_{ki}^{0} = {{\sum\limits_{q = k}^{k - 5}\quad{j^{q} \cdot b_{q} \cdot h_{{4{({k - q})}} + i}}} = {x_{0} + {j\quad y_{0}}}}} & (4) \\        {E_{ki}^{1} = {{\sum\limits_{q = k}^{k - 5}\quad{j^{q} \cdot \frac{1}{8} \cdot B_{q}^{1} \cdot h_{{4{({k - q})}} + i}}} = {x_{1} + {j\quad y_{1}}}}} & (5) \\        {E_{ki}^{2} = {{\sum\limits_{q = k}^{k - 5}\quad{j^{q} \cdot \frac{1}{64} \cdot B_{q}^{2} \cdot h_{{4{({k - q})}} + i}}} = {x_{2} + {j\quad y_{2}}}}} & (6)        \end{matrix}$

The numbers x₀, y₀, y₁, x₂ and y₂ are real numbers.

For example, the second module includes a first sampling circuit E_(o)which receives the bit b_(k) and supplies it to a first router A₀synchronized with the sampling circuit in response to a samplingfrequency Fs generated by circuit 11. The first router produces as itsoutput signal I₀, successively, the first sample of the bit b_(k) andthen the third sample of the same bit b_(k) with the sign changed. Italso produces as its output signal Q₀, successively, the second sampleof the bit b_(k) and then the fourth sample of the same bit b_(k) withthe sign changed. The second module then correlates the output signal I₀with the modulation function h according to equation (4) to produce thefirst real component x₀ (this is symbolized by the operator * in thefigure). Note that only the terms corresponding to an even index q arenon-zero.

The discrete correlation operation need not be described in more detailbecause this technique is well-known to the skilled person.

The second module also correlates the output signal Q₀ with themodulation function h according to equation (4) to produce the firstimaginary component y₀. Note that only the terms corresponding to an oddindex q are non-zero.

Similarly, the second module comprises a second sampling circuit E₁which receives the signal B_(k) ¹ and feeds it to a second router A₁synchronized with that sampling circuit. The second router produces asits output signal I₁, successively, the first sample of the term B_(k) ¹and then the third sample of that same term with the sign changed. Italso produces as its output signal Q₁, successively, the second sampleof the term B_(k) ¹ and then the fourth sample of that same term withthe sign changed. The second module then correlates the output signal I₁with the modulation function h multiplied by the constant a (⅛ in thisinstance) according to equation (5) to produce the second real componentx₁.

The second module also correlates the output Q₁ with the modulationfunction h multiplied by ⅛ according to equation (5) to produce thesecond imaginary component y₁.

The second module produces the third real component x₂ and the thirdimaginary component y₂ from the expression B_(k) ² according to equation(6) in an analogous manner.

The real part I of the modulated signal is the result of summing thethree real components x₀, x₁, x₂ and its imaginary part Q is the resultof summing the three imaginary components y₀, y₁ and y₂.

The invention naturally also relates to a demodulator for recovering thedata signal from the modulated signal S. Although the implementation ofa demodulator as specified will be evident to the skilled person, one ofmany examples of its implementation will now be described.

Referring to FIG. 4, the demodulator includes a baseband transpositionunit 12 which receives a signal r(t) modulated as described above. Thetransposition unit is routinely implemented by means of a Hilbertfilter.

The demodulator also includes a complex multiplier 13 for multiplyingthe output signal of the transposition unit 12 by the expression$e^{{- j}\frac{\pi\tau}{2T}}$to produce a signal whose frequency is equal to one-quarter of the bittime.

It also includes a convolution operator 14 which convolutes the outputsignal of the complex multiplier 13 and the modulation function h(t)defined above.

The result of this convolution is interpreted by a decision unit 15which reproduces the bit b_(k) according to the sign of the real part ofthe result.

The invention therefore relates to a digital modulation technique whichapplies regardless of how the modulation function is represented,including by means of a compression law. It is not limited to theembodiments described above. In particular, it is possible to replaceany means by equivalent means.

1. A method of modulating a digital data signal formed of a series ofbits each identified by a respective rank k and each having a durationT, the method comprising the steps of: providing an input signal F(k) asa function of said data signal; for each of a plurality of said ranks k,computing a respective product of j^(k), of a modulation functionh(t−kT) being a Gaussian function of time t and of said input signalF(k), where j is a complex constant such that j⁻²=−1; defining saidmodulation function as:h(t)=1/(σT√2Π)*e ⁻(t ²/(2σ² T ²)) where σ is a form factor determining abit spreading; and summing said computed products to provide atransmission signal.
 2. A method as claimed in claim 1, wherein the stepof providing the input signal F(k) comprises taking said input signalF(k) as equal to said data signal.
 3. A method as claimed in claim 1,wherein the step of providing the input signal F(k) comprises takingsaid input signal F(k) as equal to:${F(k)} = {\sum\limits_{n = 0}^{N}\quad{a^{n}B_{k}^{n}}}$ where: N is apositive natural integer, a is a positive correction constant, B_(k)^(n) is a polynomial defined as follows:$B_{k}^{n} = {\sum\limits_{l = 1}^{L}\quad\left( {\prod\limits_{i = 0}^{2\quad M}\quad b_{k + p_{1,i}}} \right)}$where: L is a natural integer; b_(k) is a bit of rank k M is a naturalinteger and, for each integer l such that 1≦l≦L, the p_(l,j)'s arerelative integers such that:${\sum\limits_{i = 0}^{M}\quad p_{1,{2i}}} = {\sum\limits_{i = 1}^{M}\quad p_{1,{{2i} - 1}}}$${{\sum\limits_{i = 0}^{M}\quad p_{1,{2i}}^{2}} - {\sum\limits_{i = 1}^{M}\quad p_{1,{{2i} - 1}}^{2}}} = {2n}$and p_(l,i)<p_(l,i+1) regardless of i.
 4. A method as claimed in claim3, wherein said correction constant a has the value${\mathbb{e}}^{- \frac{1}{\sigma^{2}}}.$
 5. A demodulator receiving atransmission signal obtained by the method claimed in claim 1 andreproducing said data signal, the demodulator comprising a basebandtransposition unit for receiving a signal r(t) which has been modulatedby said modulation function h(t−kT), a complex multiplier formultiplying an output signal of the transposition unit by the expression${\mathbb{e}}^{{- j}\frac{\pi\quad t}{2T}},$ a convolution operator forconvoluting an output signal of the complex multiplier and saidmodulation function, and a decision unit for reproducing said datasignal as a function of the sign of the real part of a result of saidconvolution.
 6. A modulator for modulating a digital data signal formedof a series of bits each identified by a respective rank k and eachhaving a duration T, the modulator comprising: means for providing aninput signal F(k) as a function of said data signal; computing means forcomputing, for each of a plurality of said ranks k, a respective productof j^(k), of a modulation function h(t−kT) being a Gaussian function oftime t and of said input signal F(k), where j is a complex constant suchthat j²=−1, and for summing said computed products to provide atransmission signal; wherein said modulation function is defined as:h(t)=1/(σT _(√2)Π)*e ⁻(t ²/(2σ² T ²)) where σ is a form factordetermining a bit spreading.
 7. A modulator according to claim 6,wherein the computing means comprise a digital processor for receivingsaid input signal F(k) and producing a real part and an imaginary partof said transmission signal, the modulator further comprising a firstmixer for multiplying said real part by a carrier, a phase-shifter forreceiving and phase-shifting the carrier by π/2, a second mixer formultiplying said imaginary part by an output signal of thephase-shifter, and an adder for summing output signals of the first andsecond mixers.
 8. A modulator according to claim 6, wherein the meansfor providing the input signal F(k) are arranged to take said inputsignal F(k) as equal to:${F(k)} = {\sum\limits_{n = 0}^{N}\quad{a^{n}B_{k}^{n}}}$ where: N is apositive natural integer, a is a positive correction constant, B_(k)^(n) is a polynomial defined as follows:$B_{k}^{n} = {\sum\limits_{l = 1}^{L}\quad\left( {\prod\limits_{i = 0}^{2\quad M}\quad b_{k + p_{1,i}}} \right)}$where: L is a natural integer; b_(k) is a but of rank k M is a naturalinteger and, for each integer l such that 1≦l≦L the p_(l,i)'s arerelative integers such that:${\sum\limits_{i = 0}^{M}\quad p_{1,{2i}}} = {\sum\limits_{i = 1}^{M}\quad p_{1,{{2i} - 1}}}$${{\sum\limits_{i = 0}^{M}\quad p_{1,{2i}}^{2}} - {\sum\limits_{i = 1}^{M}\quad p_{1,{{2i} - 1}}^{2}}} = {2n}$and p_(l,i)<p_(l,i+1) regardless of i).
 9. A modulator according toclaim 8, wherein the computing means comprise a digital processor forreceiving said input signal F(k) and producing a real part and animaginary part of said transmission signal, the modulator furthercomprising a first mixer for multiplying said real part by a carrier, aphase-shifter for receiving and phase-shifting said carrier by π/2, asecond mixer for multiplying said imaginary part by an output signal ofthe phase-shifter, and an adder for summing output signals of the firstand second mixers.
 10. A modulator according to claim 9, wherein saiddigital processor comprises a first module for producing saidpolynomials B_(k) ^(n).
 11. A modulator according to claim 10, whereinsaid digital processor comprises a second module for producing digitalsamples of said transmission signal, four samples E_(4k+i) beingassociated with the bit b_(k) of rank k for i varying from 0 to 3 andeach sample E_(4k+i) having the value:$E_{{4k} + i} = {\sum\limits_{q = k}^{k - 5}\quad{j^{q} \cdot \left( {b_{q} + {\frac{1}{8} \cdot B_{q}^{1}} + {\frac{1}{64} \cdot B_{q}^{2}}} \right) \cdot {h_{{4{({k - q})}} + i}.}}}$12. A demodulator according to claim 5, wherein said transposition unitcomprises a Hilbert filter.